According to [1]} the complex representation theories of the Lie groups \(\mathrm {GL}(m,\mathbb {H})\subset \mathrm {GL}(2m,\mathbb {C})\)  and \(\mathrm {U}(2m)\)  are equivalent. In particular, irreducible complex representations of \(\mathrm {GL}(m,\mathbb {H})\)  may be identified with highest weights for \(\mathrm {U}(2m)\) . For each \(k\ge 0\)  put
\(\mathbb {W}_{k}^{j} = S^{j+k}\mathbb {E}\otimes (\Lambda ^{j}\mathbb {F}\otimes S^{k}\mathbb {F}^{*})_{0} \quad \text{for } j<2m, \quad \mathbb {W}_{k}^{2m} = S^{2(m+k)}\mathbb {E}\otimes \Lambda ^{2m}\mathbb {F},\) 
